Was it Peter or Paul who first revealed that the gospel would be announced to the gentiles?
It was neither Peter or Paul that first revealed the gospel to the gentiles. Following the persecution that took place over Stephen being stoned to to death,Christians were scattered and went as far as Phoenicia, Cyprus, and Antioch (Acts 11:19-20)
Acts 11:19-20 (NET Bible)
Activity in the Church at Antioch
19 "Now those who had been scattered because of the persecution that
took place over Stephen went as far as Phoenicia, Cyprus, and Antioch,
speaking the message to no one but Jews. 20 But there were some men
from Cyprus and Cyrene among them who came to Antioch and began to
speak to the Greeks too, proclaiming the good news of the Lord Jesus."
When the Apostles and the Elders in Jerusalem heard that many Greek-speaking people were becoming believers, they dispatched Barnabas as far as Antioch, and when he observed the great interest shown there, he brought Paul in from Tarsus to help. (Acts 11:21-26)
Acts 11:20-26 (NET Bible)
21 "The hand of the Lord was with them, and a great number who
believed turned to the Lord. 22 A report about them came to the
attention[p] of the church in Jerusalem, and they sent Barnabas to
Antioch. 23 When he came and saw the grace of God, he rejoiced and
encouraged them all to remain true to the Lord with devoted hearts, 24
because he was a good man, full of the Holy Spirit and of faith, and a
significant number of people were brought to the Lord. 25 Then
Barnabas departed for Tarsus to look for Saul, 26 and when he found
him, he brought him to Antioch. So for a whole year Barnabas and Saul
met with the church and taught a significant number of people. Now it
was in Antioch that the disciples were first called Christians."
Paul (Saul of Tarsus)
Paul was chosen by Jesus to be his representative to the nations and the sons of Israel. He was a murderous opponent of Christ’s followers, but his life was changed completely when he was shown mercy and became the “chosen vessel” of the Lord Jesus Christ.
Acts 9:15 (NASB)
15 "But the Lord said to him, “Go, for he is a chosen [a]instrument of
Mine, to bear My name before the Gentiles and kings and the sons of
1 Timothy 1:12-13 (NASB)
12 "I thank Christ Jesus our Lord, who has strengthened me, because He
considered me faithful, putting me into service, 13 even though I was
formerly a blasphemer and a persecutor and a violent aggressor. Yet I
was shown mercy because I acted ignorantly in unbelief;"
Peter also known in the scriptures as Cephas- Symeon-Simon)
Peter was privileged to open the hope of entering God’s Kingdom to three different fields of mankind, first to the Jews, then to the Samaritans, and finally to the Gentiles, or non-Jews.
Matthew 16:19 NIV
19 "I will give you the keys of the kingdom of heaven; whatever you
bind on earth will be bound in heaven, and whatever you loose on earth
will be[e] loosed in heaven."
Peter alone was given the keys of the kingdom. He used these "keys" entrusted to him to open up ,for Jews Acts 2:22-41, for Samaritans Acts 8: 14-17, for the Gentiles -Cornelius. Acts 10: 1-10 , 10:34-38 , the opportunity to receive God’s spirit with a view to their entering the heavenly Kingdom.